Zax

"You are locked in a 50x50x50-foot room which sits on 100-foot stilts. There is an open window at the corner of the room, near the floor, with a strong hook cemented into the floor by the window. So if you had a 100-foot rope, you could tie one end to the hook, and climb down the rope to freedom. (The stilts are not accessible from the window.) There are two 50-foot lengths of rope, each cemented into the ceiling, about 1 foot apart, near the center of the ceiling. You are a strong, agile rope climber, good at tying knots, and you have a sharp knife. You have no other tools (not even clothes). The rope is strong enough to hold your weight, but not if it is cut lengthwise. You can survive a fall of no more than 10 feet. How do you get out alive?"

Given Euler's identity [ 1 ] , e^( iπ ) = -1 , and

an Eisenstein integer [ 2 ] , w = e^( 2πi/3 ) ,

shouldn't ( e^( iπ ) )^( 2/3 ) = w ?

But does -1^( 2/3 ) = ½ ( -1 + √3 ) ?

Also , does w^( 3/2 ) = e^( 2πi/3 )^( 3/2 ) = -1 ?

[1] http://en.wikipedia.org/wiki/Euler%27s_identity

[2] http://en.wikipedia.org/wiki/Eisenstein_integers

4ˣ - 2ˣ - 1 = 0

aˣ = x + b

Given

( x + a ) ² + y ² = r ² , ( intersects y axis at ±4 )

( x₁ - a₁ ) ² + y₁² = r₁²

( x₂ + a₂ ) ² + y₂² = r₂²

r₁ + r₂ = r

The y axis is tangent to A₁ and A₂ .

Find the enclosed area A - A₁ - A₂

33x + 55y + 79z - 99w = r

The solution requires x , y , z , and w

to be expressed in terms of r , s , t , u .

Given :

x² + y² - 18y = 0

x² - 24x + y² - 8y + 144 = 0

x² - 14.4x + y² - 2.88y + 51.84 = 0

How are the radii related ?